[curves] Moving On... What about knots?

Brad Klee bradklee at gmail.com
Mon Apr 5 19:31:17 PDT 2021

Arg, technical difficulties. Here are the promised curves
(for humour value / general interest if nothing else) :


Each knot determines a cyclic group, and the tent map
iterates between intersections with blue boundaries.

The tent map on rational inputs can be used in DLP.
Are tent map cyclic groups "too easy" or just "too slow"?
Is there a sensible way to define pairing for two equal-length
tent map cycles?

Another option is to hook up a whole lattice of interlinked
families of Edwards curves, then choose an irrational input
such as Pi. The trajectory will take a randomish walk through
the plane, and the endpoints can be used as public keys.
How bad an idea is this in terms of attacks?

(I know, the ECC addition rules are fast, especially due
to point doubling, but continue to explore what else can
be done.)

This idea, partially from Joan Birman, seems to have
some promise, so I will try to write more about it when
conditions are more conducive to work.

Hope that soon the pandemic ends and we can achieve
world peace, shalom/salama, kosen rufu, etc.


On Sun, Apr 4, 2021 at 11:37 AM Brad Klee <bradklee at gmail.com> wrote:

> This also doesn't make sense... when you answer a question you ask
> by talking to yourself online (not as a joke), see again:
> https://community.wolfram.com/groups/-/m/t/2232630
> (with added Voltaire translation & hashclash link.)
> Where is the hidden answer and why? Here: https://0x0.st/-ci5.text .
> Not too off-topic in my opinion, but for "semi-purists" to get back within
> list parameters ("New Curves"), see also Test 1 & Test 2:
> https://community.wolfram.com/groups/-/m/t/2236396
> https://community.wolfram.com/groups/-/m/t/2237003
> Not sure how knot recognition would be used for cryptography, but the
> problem certainly meets a difficulty requirement. Interesting article
> from Sergei Gukov et al. "Learning to Unknot":
> https://arxiv.org/abs/2010.16263
> I didn't see if Aurore has anything about machine learning, so far I don't,
> but it looks like homology & security will both rely increasingly on
> smart,
> self-evolving techniques.
> --Brad
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